Key polynomials over valued fields

Abstract

Let $K$ be a field. For any valuation $\mu$ on $K[x]$ admitting key polynomials we determine the structure of the whole set of key polynomials in terms of a fixed key polynomial of minimal degree. We deduce a canonical bijection between the set of $\mu$-equivalence classes of key polynomials and the maximal spectrum of the subring of elements of degree zero in the graded algebra of $\mu$.